Creating an Unwrapping Animation (NOT texturing)
I'm not quite sure what you are trying to achieve, but I created this blend file that turns a 2D fan into a 3D cone... https://www.dropbox.com/s/zjddbv90rfj48n8/Cone2Dto3D.blend?dl=0 . Just press Alt+A to run the animation.

OK, since it was requested, here's the instructions....

0) To get started, Num Pad 7 for top view and Shift+C to centre cursor at origin.

1) Add a mesh circle with 72 sides, triangle fan fill, 1.1536m radius. [Any strange dimensions given here are due to the maths required to make a cone that ends up as 1m tall. I did do the maths but deleted it].

2) R, Z, 2.5, Enter, to rotate circle about Z by 2.5 degrees.

3) Tab into edit mode and delete 71 of the faces, leaving just the one at the 12 o'clock position.

4) Add an Empty ('Empty1') at the origin (i.e. the sharp point of the triangle).

5) Select just the triangle and add an array modifier, Count=36. Turn off Relative Offset and choose just Object Offset, specifying Empty1 as the Object Offset.

6) Rotate Empty1 5 degrees about Z to cause a fanned array of triangles in a semi-circle.

7) Add another Empty ('Empty2') at the point of the triangle (I used the Single Arrow style to make things clearer). You should get an arrow pointing up the +ve Z axis.

8) Select Empty2, then shift-select the triangle, hit Tab to edit triangle, shift select the vertex at the point.

9) Press Ctrl+H, Hook to Selected Object. (Moving Empty2 can now drag the triangles up and down into a point... but leave it where it is for now at Z=0).

10) With the triangle still in edit mode, select the other two vertices only (Ctrl+I), Shift+S, Cursor To Selected. The cursor should now be between the two vertices at the outside of the triangle.

11) Tab out of edit mode and then A to deselect the triangles.

12) Add another Empty ('Empty3'), single arrow style, then press R, X, 90, Enter, so it points back towards the point of the triangle (along the Y axis).

13) With Empty3 selected, shift-select the triangle, Tab, Ctrl+H, Hook To Selected Object. [Moving Empty3 can now squash the triangle towards the centre... but leave it where it is for now at Y = 1.1536m.]

14) Tab out of edit mode and A to deselect all.

++ Now the animation ++

15) Set start frame to 1, end frame to anything you like.

16) At Frame 1, select Empty2 (the arrow pointing upwards) and set a key frame for Loc Z (i.e. Z=0).

17) Go to last frame (Shift+Right arrow). Move Empty2 up to Z=1 and set a keyframe.

18) IMPORTANT! Go into the graph editor and set interpolation (T) to Linear.

19) Go back to Frame 1 (Shift+Left arrow), select Empty1 and set a keyframe for Rot Z (i.e. Rot Z = 5 degrees).

20) Go to last frame. Rotate Empty2 another 5 degrees about Z and set a keyframe (i.e. Rot Z = 10 degrees). [You should have a droopy propellor / palm leaf thing now].

21) IMPORTANT! Go into the graph editor and set interpolation (T) to Linear.

22) Go back to Frame 1, select Empty3 and set a keyframe for Loc Y (i.e. Y = 1.1536)

23) Go to last frame. In top view, select Empty3 and set its Y location to Y = 0.57625. The triangles should join exactly to make a cone. If not, move Empty3 manually along Y until they do. 

24) When the triangles meet exactly, set a keyframe for Empty3's Y location.

25) IMPORTANT! Go into the graph editor, select Empty3's graph and open the Properties panel (P). Set Interpolation to Sinusoidal, and set Easing to Ease Out.

26) Alt+A to run animation.